Q.64 - The Practice of Statistics for AP

Question

 Bad carpet The number of flaws per square yard is a type of carpet material that varies with mean $1.6$ flaws per square yard and standard deviation $1.2$ flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies $200$ square yards of the material records the number of flaws found in each square yard, and calculates $x$, the mean number of flaws per square yard inspected. Find the probability that the mean number of flaws exceeds $2$ per square yard. Follow the four-step process. 

Step-by-Step Solution

Answer

From the given information, the required probability is $0.0001$

Step 1: Given Information

It is given in the question that, the mean,$\mu =1.6$

the standard deviation, $\sigma =1.2$

the number of materials studied = $200$ square yards.

Find the probability that the mean number of flaws exceeds $2$ per square yard. 

Step 2: Explanation

The central limit theorem states that if the sample size of a sampling distribution is $30$ or more, then the sample mean is approximately normal whose mean is $\mu$, and the standard deviation is  $\frac{\sigma }{\sqrt{n}}$.

The z-value of a distribution can be found by dividing the difference between the population mean and sample mean by the standard deviation that is, $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$.

Step 3: Explanation

The sample size of 200 square yards is at least 30; so apply the central limit theorem.

Find the z value by using the formula $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$.

Substitute $2 for x, 1.6 for \mu ,1.2 for \sigma and 200 for n$in the above formula and simplify.

$z=\frac{2-1.6}{\frac{1.2}{\sqrt{200}}}$

 $=\frac{0.4}{1.2}\times \sqrt{200}$

 $=4.71$

Thus, the corresponding probability is:

$P(\overline{x}>2)=P(z>4.71)=P(Z<-4.71)=0.0001$

Hence, the required probability is $0.0001$